Chattering Reduction Techniques in Sliding Modes
Control Implemented in Discrete Time
M´onica E. Romero
Depto. de Electr´onica, Universidad Nacional de Rosario
Riobamba 245 bis, (2000) Rosario, Argentina,
During her academic visit period in :
LSIS, Laboratoire des Sciences de l’ Informatiion et des Syst`eme
Ecole Politechnique de Marseille, France
Email: mromero@fceia.unr.edu.a
Abstract—This paper presents two strategies for chattering
reduction on Sliding Mode controlled systems. In this case the
controller is designed using Lyapunov approach wich provides a
powerfull tool working with non-linear multi-input multi-output,
(MIMO). The implementation of the controller is in discrete
time. Sampling time effects are taken into account to propose
the reduction techniques. Simulation results are shown in order
to demonstrate de viability of the method presented here.
I. INTRODUCTION
One of the main drawback of the Sliding Mode Control
(SMC) technique is the chattering phenomenon, i. e., the
oscillatory motion of the controlled system about the sliding
manifold, [1].
There are two possible mechanisms which produces such a
motion: a.- presence of parasitic dynamics in series with the
plant and 2.-switching nonidealities [1].
Many works in the literature are devoted to caracterize
the chattering in magnitud and frequency as well as the
implementation of chattering reduction technique, [2],[3],[4].
In this article, we present a chattering reduction technique
for a SM controller designed with the Lyapunov approach
and implemented in discrete time. The analysis is done when
the system is working in sliding mode, i.e. in steady state
condition, avoiding the transitory regime. The only constrain
for the reduction results effective is the open loop system
should be stable. This feature could be interpreted as a very
restrictive one, but, in the field of the power electronic devices,
for example electrical machinery, the stability of the open loop
system is not a great deal. In this particular application field,
where SMC is widely applied, ( [5], [6], [7], [8], [9]) the
actuators are power electronic converters that include switches
networks . As the actuator reproduces the control signal as
a PWM signals, the chattering phenomenon is a inherent
characteristic of the system nature instead of been an aditional
disadvantage.
As we already mentioned we deal with SMC implemented
in discrete time. The sampling time Ts is fixed taking into
account the time response of the actuator, the control algorithm,
the analog-digital convertions and, if there exist, the
unmodeled dynamics.
The organization of the paper is as follow, section §II
presents the basic control strategy, section §III presents a
chattering characterization, in section § IV we present the
reduction ripple techniques, in section §V some simulation
results are presented, finally section § VI presents a robustness
analysis and section §VII summarize the conclusions.
II. SLIDING MODE CONTROL. DEFINITION AND CONTROL
DESIGN
Consider the system
? x = f(x) + g(x)u (1)
where x ? X ? n is the state vector, u ? m is the control
vector, and f y g are infinity diferenciable vector fields. We
note the columns of g(x) as g1(x), g2(x), . . . , gm(x).
Let define a vectorial conmutation funtion S T =
[S1S2 . . . Sm], and the control target is to achieve S = 0 with
a control vector u such that each componets can takes the
value u+ or u-, with u+ = u-.
A. Basic Strategy
There are many ways to design the controller with sliding
mode approach for MIMO systems, depending on the
switching scheme ([10]). We will use here the so call Eventual
Sliding Switching Scheme, and the appropriated technique for
the control design is the Lyapunov approach.
For system (1) we propose a Lyapunov function candidate:
W =
1
2STS > 0 (2)
. To achieve the convergence to the manifold S = 0, the
condition dW
dt < 0 should be accomplished.
The time derivative of W along the system trajectories
results:
dW
dt
= ST dS
dt
= ST (H + Du), (3)
where
H 
 ?S
?x
f(x) + ?S
?t
(x) (4)
and D 
 ?S
?x
g(x) + ?S
?t
(u) (5)
.....